Mesh does not follow armature
you used NGons the bad way.

  

Everything is fine after triangulation of the faces (Edit mode Tab, select all A, Ctrl+F and "triangulate faces" in the face menu or Ctrl+T)

Recalculate normals (all selected and Ctrl+N)
Back in object mode Tab, select the mesh RClick, then the armature with Shift+RClick, then Ctrl+P and choose "armature deform"/"with automatic weights"
Then parenting the hand to the armatures works fine.